Question 14 in F-I-S section 1.2 asks:
Let $\mathbf{V}=\{(a_1,a_2,\ldots ,a_n)\colon a_i\in \mathbb{C}$ for $i=1,2,\ldots n\}$; so $\mathbf{V}$ is a vector space over $\mathbb{C}$. Is $\mathbf{V}$ a vector space over the field of real numbers with the operations of coordinatewise addition and multiplication?
My question concerns which field operators are relevant when considering $\mathbf{V}$ over $\mathbb{R}$. I'm confused because if we take $x=(a+bi)$ and $y=(c+di)$ where $b,d\neq 0$ so $x,y\in\mathbf{V}$, then being "over the field of $\mathbb{R}$" would be create a conflict when considering $x+y$ (the reasoning being that complex addition is not defined for the real numbers).
Even with scalar multiplication, if we multiply $x\in\mathbf{V}$ by $a\in \mathbb{R}$, then I'm still using scalar multiplication as defined in the field $\mathbb{C}$.
So why if I'm taking $\mathbf{V}$ over $\mathbb{R}$ are we still using the field operators as defined by $\mathbb{C}$?
EDIT:
Thank you for the responses! But I think I'm still confused by the fact that $\mathbf{V}$ is a set of vectors with complex components but it's being taken over $\mathbb{R}$.
So let's say I'm now checking to see if $\mathbf{V}$ is a vector space over $\mathbb{R}$. If I take this set $\mathbf{V}$ over $\mathbb{R}$, still retaining coordinatewise addition, then when I perform $(a_1,a_2,\ldots ,a_n)+(b_1,b_2,\ldots ,b_n)$ with $a_i,b_i\in\mathbb{C}$, then $a_i+b_i$ (having complex parts) isn't defined with the addition operator for $\mathbb{R}$. So in this case, how do I verify the vector space requirements such as associativity if addition (from $\mathbb{R}$) isn't defined for the vectors in $\mathbf{V}$ (those that have components with complex parts).
Or is it true to say that the addition operator from the field $\mathbb{R}$ is still defined when adding complex numbers?